A304134 Number of partitions of 5n into exactly n parts.

1, 1, 5, 19, 64, 192, 532, 1367, 3319, 7657, 16928, 36043, 74287, 148702, 290071, 552767, 1031391, 1887776, 3395084, 6007963, 10474462, 18010859, 30574655, 51284587, 85064661, 139620591, 226914505, 365371100, 583164222, 923075291, 1449643115, 2259616844

Offset: 0

Author

Seiichi Manyama, May 07 2018

Keywords

• nonn

Comments

Also, the number of partitions of 4n in which every part is <=n.

Examples

			n | Partitions of 5n into exactly n parts
--+------------------------------------------------
1 | 5;
2 | 9+1, 8+2, 7+3, 6+4, 5+5;
3 | 13+1+1, 12+2+1, 11+3+1, 11+2+2, 10+4+1, 10+3+2,
  |  9+5+1,  9+4+2,  9+3+3,  8+6+1,  8+5+2,  8+4+3,
  |  7+7+1,  7+6+2,  7+5+3,  7+4+4,  6+6+3,  6+5+4,
  |  5+5+5;
====================================================================
n | Partitions of 4n in which every part is <=n.
--+-----------------------------------------------------------------
1 | 1+1+1+1;
2 | 2+2+2+2, 2+2+2+1+1, 2+2+1+1+1+1, 2+1+1+1+1+1+1, 1+1+1+1+1+1+1+1;
3 | 3+3+3+3, 3+3+3+2+1, 3+3+3+1+1+1, 3+3+2+2+2, 3+3+2+2+1+1,
  | 3+3+2+1+1+1+1, 3+3+1+1+1+1+1+1, 3+2+2+2+2+1, 3+2+2+2+1+1+1,
  | 3+2+2+1+1+1+1+1, 3+2+1+1+1+1+1+1+1, 3+1+1+1+1+1+1+1+1+1,
  | 2+2+2+2+2+2, 2+2+2+2+2+1+1, 2+2+2+2+1+1+1+1, 2+2+2+1+1+1+1+1+1,
  | 2+2+1+1+1+1+1+1+1+1, 2+1+1+1+1+1+1+1+1+1+1,
  | 1+1+1+1+1+1+1+1+1+1+1+1;
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..3000 (first 501 terms from Seiichi Manyama)

Crossrefs

Cf. A209816, A209818.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
      b(n, i-1) +b(n-i, min(i, n-i)))
    end:
a:= n-> b(4*n, n):
seq(a(n), n=0..35);  # Alois P. Heinz, May 07 2018

Mathematica

b[n_, i_] := b[n, i] = If[n==0 || i==1, 1, b[n, i-1] + b[n-i, Min[i, n-i]]];
a[n_] := b[4n, n];
a /@ Range[0, 35] (* Jean-François Alcover, Nov 27 2020, after Alois P. Heinz *)

PARI

{a(n) = polcoeff(prod(k=1, n, 1/(1-x^k+x*O(x^(4*n)))), 4*n)}

Extensions

More terms from Alois P. Heinz, May 07 2018